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\null \vskip 26truemm
\noindent{\title Quantum Plasma Model with Hydrodynamical}
\vskip 0truemm
\noindent{\title Phase Transition}
\vskip 12 truemm
\noindent {{\auth By Geoffrey L. Sewell}\footnote *{Partially
supported by European Capital and Mobility Contract No. CHRX-CT-
0007}}
\vskip 5truemm
\noindent Department of Physics, Queen Mary and Westfield
College, 
\hfil\break
Mile End Road, London E1 4NS
\vskip 26truemm
\noindent {\abs {\titabs Abstract.}
We derive the electro-hydrodynamics of the Jellium plasma model
from its many-particle Schr\"odinger equation, subject to certain
general initial and regularity conditions, and prove that it
undergoes a transition from deterministic to stochastic flow when
a certain parameter, representing the non-uniformity in the
initial density and drift velocity profiles, reaches a certain
critical value. Thus, the model exhibits a phase transition far
from equilibrium.}
\vskip 12truemm 
\noindent {\subt 1 Introduction}
\medskip\noindent
The quantum Jellium model is a system of electrons, interacting
via Coulomb forces both with one another and with a uniform,
positively charged, neutralising background. It is thus a model
of a many-particle system with realistic interactions. At the
level of mathematical physics, it has been proved to enjoy 'good'
thermodynamic [1,2] and hydrodynamic [3] properties. In fact,
apart from Davies's [4] derivation of Fourier's law of heat
conduction for a certain model of interacting atoms, the passage
from quantum mechanics to Eulerian hydrodynamics in [3]
represents, to the best of our knowledge, the only rigorous
quantum statistical derivation of a macroscopic continuum
mechanics. It is, however, based on the assumption of regularity
conditions, which exclude the possibility of hydrodynamical phase
transitions.
\vskip 0.2cm\noindent
The object of the present article is to provide a further quantum
mechanical treatment of the hydrodynamics of the Jellium model,
in which certain regularity assumptions of [3] are weakened in
such a way as to admit non-equilibrium phase transitions. In
fact, we show that, under the new assumptions, the model exhibits
a transition from a deterministic to a stochastic hydrodynamics
when a certain parameter, representing the non-uniformity of the
initial density and velocity profiles, attains a critical value. 
The method by which we obtain this result is based on two
main steps. Firstly, we derive a Vlasov equation for the large
scale dynamics of the model from its many-particle Schr\"odinger
equation, subject to specified initial and regularity conditions.
We then show that the Vlasov dynamics reduces to a deterministic
Eulerian hydrodynamics if the initial density and velocity
profiles lie below a certain non-uniformity threshold, and that
otherwise the flow becomes stochastic. Specifically, in the
latter case, the flow corresponds to a {\it statistical mixture}
of different streams, and thus the local density and drift
velocity have macroscopic dispersions.
\vskip 0.2cm\noindent
We present our treatment as follows. In ${\S}$2, we extend
the scheme of Refs. [3,5] so as to derive a Vlasov equation,
governing the large-scale dynamics of the Jellium model, from its
many-particle Schr\"odinger equation, subject to rather
general initial conditions.  To be more precise, we provide a
treatment here of both the Jellium model itself and a regularised
version of it, obtained by introducing a short distance cut-off
in the Coulomb potential. For the regular model, our derivation
of the Vlasov dynamics is based exclusively on the Schr\"odinger
equation and the initial conditions. For the true Jellium model,
on the other hand, certain supplementary regularity assumptions
are also required. The essential idea behind these is that the 
repulsive character of the inter-electronic forces keeps the 
electrons apart and thereby tames the singularity in the Coulomb 
potential. The precise form of the assumptions is specified by 
the conditions (R.1-4). We remark here that we have not avoided 
repeating much of the formalism of Ref. [3] in this Section, 
because it is essential for our present purposes to reset it 
in the context of our new (weakened) regularity conditions
\vskip 0.2cm\noindent
In ${\S}$3, we note that the Vlasov equation is just the
Liouville equation for a certain {\it Lagrangian}
hydrodynamics, governing the evolution of the time (t)-dependent
position, $X_{t}(x),$ of a 'fluid particle'
located initially at the point $x.$  We then show that the Vlasov
dynamics reduces to a deterministic Eulerian hydrodynamics if and
only if the function $X_{t}$ is invertible: otherwise it
corresponds a stochastic flow, in the sense described above.
\vskip 0.2cm\noindent
In ${\S}$4, we analyse the conditions for deterministic
versus stochastic flow in both the true Jellium model and the
regularised one, in the situation where the initial density and
velocity profiles depend on only one spatial coordinate and so
are essentially one-dimensional. For this case, we are able to
show explicitly that both models undergo transitions from
deterministic to stochastic flow when the initial conditions
attain certain non-uniformity thresholds.
\vskip 0.2cm\noindent
We conclude, in ${\S}$5, with some brief further comments
on the results obtained here and on their possible relevance to
the theory of turbulence.
\vfill\eject
\noindent {\subt 2. Basis of the Vlasov Dynamics}
\medskip\noindent
The Jellium model, ${\Sigma}^{(N,L)},$ consists of $N$ electrons
in a cube, $K^{(L)},$ of side $L,$ with uniform neutralising
positive charge background. We assume periodic boundary
conditions. Our objective will be to obtain a quantum theoretical
derivation of the hydrodynamics of ${\Sigma}^{(N,L)}$ in a limit
where $N$ and $L$ tend to infinity and the mean particle density,
$${\overline n}=N/L^{3}\eqno(2.1)$$
remains fixed and finite. 
\vskip 0.3cm \noindent
We denote the position vectors and momenta of the electrons by
$X_{1},.. \  .,X_{N}$ and $P_{1},.. \ .,P_{N},$ respectively.
Thus, $P_{j}=-i{\hbar}{\nabla}_{j}^{(L)},$ where 
${\nabla}^{(L)}$
is the gradient operator in $K^{(L)}.$  At the microscopic
level, the pure states of the system are given by the normalised,
antisymmetric wave-functions ${\Psi}^{(N)}(X_{1},.. \ .,X_{N}),$
and the Hamiltonian takes the form
$$H^{(N,L)}={-{\hbar}^{2}\over 2m}{\sum}_{j=1}^{N}
{\Delta}_{j}^{(L)}+e^{2}{\sum}_{j,k(>j)=1}^{N}U^{(L)}(X_{j}-X_
{k})
\eqno(2.2)$$
where $-e,m$ are the electronic charge and mass, respectively,
${\Delta}^{(L)}$ is the Laplacian for $K^{(L)},$ and $U^{(L)}(X)$
is the difference between ${\vert}X{\vert}^{-1},$ periodicised
w.r.t. $K^{(L)},$ and its space average over that cube, i.e.
$$U^{(L)}(X)={4{\pi}\over L^{3}}{\sum}^{(L)}
{{\exp}(iQ.X)\over Q^{2}}\eqno(2.3)$$
the superscript $(L)$ over ${\Sigma}$ signifying that summation
is taken over the non-zero vectors $Q=(2{\pi}/L)(n_{1},n_{2},
n_{3}),$ with the $n$'s integers. The time-dependent
Schr\"odinger equation for ${\Sigma}^{(N,L)},$ with $T$ the time
variable, is
$$i{\hbar}{{\partial}{\Psi}_{T}^{(N)}\over {\partial}T}=
H^{(N,L)}{\Psi}_{T}^{(N)}\eqno(2.4)$$
We shall assume the following initial kinetic and potential
energy bounds for ${\Sigma}^{(N,L)}$-more precisely, for the
family of systems ${\lbrace}{\Sigma}^{(N,L)}{\rbrace},$ with
$N,L$ satisfying (2.1).
\vskip 0.2cm\noindent
$(I.1)^{(L)}$ The expectation value of the total kinetic energy
per particle, for the initial state ${\Psi}_{0}^{(N)},$ is less
than some finite $N-$independent constant $B/2m,$ i.e.
$$({\Psi}_{0}^{(N)},P_{1}^{2}{\Psi}_{0}^{(N)})<B\eqno(2.5)$$
\vskip 0.2cm\noindent
$(I.2)^{(L)}$ The expectation value of the total potential
energy, for the the initial state ${\Psi}_{0}^{(N)},$ is less
than some finite $N-$independent constant, $e^{2}C/2,$ times
$N^{5/3}.$ This bound corresponds to the electrostatic energy of
a continuous distribution of charge, whose density is a smooth
function of $X/L,$ and signifies that
$$({\Psi}_{0}^{(N)},U^{(L)}(X_{1}-X_{2}){\Psi}_{0}^{(N)})<
CN^{2/3}\eqno(2.6)$$
\vskip 0.2cm\noindent
We base our macroscopic description of the model on scales of
length, time and particle momentum given by $L,  \
{\omega}^{-1}$ and $mL{\omega},$ where ${\omega}$ is the
classical plasma frequency, i.e. 
$${\omega}=(4{\pi}{\overline n}e^{2}/m)^{1/2}\eqno(2.7)$$
For this description, we employ the rescaled space and time
coordinates, $x=X/L, \ t={\omega}T,$ respectively. Under this
rescaling, ${\Sigma}^{(N,L)}$ is mapped onto a system
${\Sigma}^{(N)}$ of particles in a unit cube,
$K,$ with periodic boundaries. Correspondingly, the state,
${\Psi}_{T}^{(N)},$ of ${\Sigma}^{(N,L)}$ is transformed to that
of ${\Sigma}^{(N)}$ given by
$${\psi}_{t}^{(N)}(x_{1},.. \ .,x_{N})=L^{3N/2}
{\Psi}_{{\omega}^{-1}t}^{(N)}(Lx_{1},.. \ .,Lx_{N})\eqno(2.8)$$
The Schr\"odinger equation (2.4) thus transforms to 
$$i{\hbar}_{N}{{\partial}{\psi}_{t}^{(N)}\over {\partial}t}=
H^{(N)}{\psi}_{t}^{(N)}\eqno(2.9)$$
where
$$H^{(N)}={1\over 2}{\sum}_{j=1}^{N}p_{j}^{2} \ + \
N^{-1}{\sum}_{j,k(>j)=1}^{N}U(x_{j}-x_{k})\eqno(2.10)$$
$$p_{j}=-i{\hbar}_{N}{\nabla}_{j}\eqno(2.11)$$
$${\hbar}_{N}={{\hbar}\over mL^{2}{\omega}}=
{{\hbar}\over m{\omega}}({{\overline n}\over
N})^{2/3}\eqno(2.12)$$
is a dimensionless effective 'Planck constant', ${\nabla}$ is the
gradient operator for $K,$ and 
$$U(x)=U_{c}(x):={\sum}_{q}^{(1)}{\exp}(iq.x)/q^{2}\eqno(2.13)$$
the superscript $(1)$ over ${\Sigma}$ signifying that summation 
is taken over the non-zero vectors $2{\pi}(n_{1},n_{2},n_{3}),$ 
with the $n$'s integers. Our reason for introducing the symbol
$U_{c}$ here is that we want to employ the Hamiltonian given by
(2.10) both for the model ${\Sigma}^{(N)},$ with $U=U_{c},$ and
for a modified version of this, where $U$ is a 'smoothed out'
Coulomb potential. We note that it follows from (2.13) that
$${\Delta}U_{c}(x)=1-{\delta}(x)\eqno(2.14)$$
where ${\Delta}$ is the Laplacian and ${\delta}$ the Dirac 
distribution for $K.$
\vskip 0.2cm\noindent
{\ssubt Note.} Two key features of the rescaled description, as
given by (2.9)-(2.14), are that
\vskip 0.2cm\noindent
(a) the effective, dimensionless Planck constant, ${\hbar}_{N},$
governing the quantum behaviour of the model, tends to zero as
$N{\rightarrow}{\infty};$ and
\vskip 0.2cm\noindent
(b) the pair interaction potential scales as $N^{-1}.$
\vskip 0.2cm\noindent
The properties (a) and (b) are generally the hallmarks of a
classical and of a mean field theory, respectively, in the limit
$N{\rightarrow}{\infty}.$ In fact, as we shall presently show,
the model does indeed reduce to a classical mean field theoretic
one, governed by Vlasov dynamics, in this limit.
\vskip 0.2cm\noindent
We formulate the dynamics of ${\Sigma}^{(N)}$  in
terms of its characteristic functions,
$${\mu}_{t}^{(N,n)}({\xi}_{1},.. \ .,{\xi}_{n};{\eta}_{1},.. \
.,{\eta}_{n})=({\psi}_{t}^{(N)},{\Pi}_{j=1}^{n}
({\exp}(i{\xi}_{j}.p_{j}/2){\exp}(i{\eta}_{j}.x_{j})
{\exp}(i{\xi}_{j}.p_{j}/2)){\psi}_{t}^{(N)})
\eqno(2.15)$$
where the ${\xi}'$s and ${\eta}'$s run over the ranges ${\bf
R}^{3}$ and $(2{\pi}{\bf Z})^{3},$ respectively. These functions
are in one-to-one correspondence with the reduced density
matrices for the state ${\psi}_{t}^{(N)};$ and, in particular,
the n-particle spatial density function   
$${\rho}_{t}^{(N,n)}(x_{1},.. \ .,x_{n})={\int}
dx_{n+1}.. \ .dx_{N}{\vert}{\psi}_{t}(x_{1},x_{2},.. \ .,x_{N})
{\vert}^{2}\eqno(2.16)$$ 
is the Fourier transform of ${\mu}_{t}^{(N,n)}$ w.r.t. the
${\eta}'$s, when the ${\xi}'$s are held at the value zero.
\vskip 0.2cm\noindent
The initial condition for ${\Sigma}^{(N)},$  corresponding to
$(I.1)^{(L)}$ for ${\Sigma}^{(N,L)},$ is
$$({\psi}_{0}^{(N)},p_{1}^{2}{\psi}_{0}^{(N)})
<N^{-2/3}b \ {\rightarrow}0 \ as \
N{\rightarrow}{\infty}\eqno(2.17)$$
with $b$ a finite constant. We shall find it useful to generalise
this condition by imposing an initial position-dependent drift
velocity, $u_{0}={\nabla}{\phi},$ on the system. This is achieved
by rephasing ${\psi}_{0}$ by the factor
${\exp}(i{\sum}_{j=1}^{N}{\phi}(x_{j})/{\hbar}_{N}),$ and results
in the replacement of (2.17) by
\vskip 0.2cm\noindent
$(I.1)$
$$({\psi}_{0}^{(N)},(p_{1}-u_{0}(x_{1}))^{2}{\psi}_{0}^{(N)})
<N^{-2/3}b{\rightarrow}0 \ as \
N{\rightarrow}{\infty}\eqno(2.18)$$
{\it  We shall assume that the function $u_{0}$ is continuously
differentiable.} 
\vskip 0.2cm\noindent
The initial condition $(II.2)^{(L)}$ transforms to the following
form for ${\Sigma}^{(N)},$ which is unaffected by the above
rephasing of the initial state.
\vskip 0.2cm\noindent
$(I.2)$
$$({\psi}_{0}^{(N)},U(x_{1}-x_{2}){\psi}_{0}^{(N)})<c
\eqno(2.19)$$
{\it where $c$ is a finite constant. We further assume that}
\vskip 0.2cm\noindent
$(I.3)$
$${\lim}_{N\to\infty}{\rho}_{0}^{(N,1)}(x)={\sigma}_{0}(x)
\ {\forall}x{\in}K\eqno(2.20)$$ 
{\it where the function ${\sigma}_{0}$ is continuous; and that}
\vskip 0.2cm\noindent
$(I.4)$
$${\lim}_{N\to\infty}({\mu}_{0}^{(N,n)}({\xi}_{1},.. \
.,{\xi}_{n};{\eta}_{1},.. \
.,{\eta}_{n})-{\Pi}_{1}^{n}{\mu}_{0}^{(N,1)}
({\xi}_{j},{\eta}_{j})){\equiv}0\eqno(2.21)$$
This last condition signifies that, in the limit
$N{\rightarrow}{\infty},$ the initial correlations of
${\Sigma}^{(N)}$ are of zero range: the assumption behind
this is that the initial state of the microscopic model
${\Sigma}^{(N,L)},$ carries only short range correlations, as in
a pure phase [6,7].
\vskip 0.2cm\noindent
We note here that it was shown by explicit construction, in the 
Appendix of Ref. [3], that the initial conditions (I.1)-(I.4) are
perfectly viable. 
\vskip 0.2cm\noindent
The macroscopic dynamics of the Jellium model, then, is
represented by the time-dependence of the characteristic
functions ${\mu}_{t}^{(N,n)},$ in the limit
$N{\rightarrow}{\infty},$ subject to the initial conditions
(I.1-4). Before attempting to extract this dynamics from the
Schr\"odinger equation (2.9), we shall first summarise results
on the corresponding problem for the simpler model,
${\Sigma}_{g}^{(N)},$ obtained by replacing the (singular)
Coulomb potential, $U_{c},$ by a suitably regular one, $U_{g},$
given by
$$U_{g}(x)={\int}_{K}dyg(x-y)U_{c}(y)\eqno(2.22)$$
where the 'smoothing function' $g,$ and hence $U_{g},$ is twice 
continuously differentiable.\footnote *{This last condition
ensures that the model meets the requirements of Refs. [5,8] 
for the derivation of the results given by (A)-(D) below.}
\vskip 0.2cm\noindent
{\ssubt The Regularised Model, ${\Sigma}_{g}^{(N)}.$} The
Hamiltonian for this model is still given by (2.10), but now with
$U=U_{g}.$ We note that this Hamiltonian has the following
simplifying features.
\vskip 0.2cm\noindent
(a) The effective Planck constant, ${\hbar}_{N},$ vanishes in the
limit $N{\rightarrow}{\infty};$ and
\vskip 0.2cm\noindent
(b) the two-body potential is a regular one, scaled by a factor
$N^{-1}.$
\vskip 0.2cm\noindent
It follows immediately from Ref. [5] that (a) and (b)
lead to a classical Vlasov dynamics. Specifically, under the
initial conditions (I.1-4), we have the following results.
\vskip 0.2cm\noindent
(A) {\it The functions ${\mu}_{t}^{(N,n)}$ converge pointwise,
as
$N{\rightarrow}{\infty},$ to the characteristic functions,
${\mu}_{t}^{(n)},$ of classical probability measures
$m_{t}^{(n)}$ on $(K{\times}{\bf R}^{3})^{n},$ i.e.}  
$${\mu}_{t}^{(n)}({\xi}_{1},.. \ .,{\xi}_{n};{\eta}_{1},.. \
.,{\eta}_{n})={\int}{\exp}({\sum}_{j=1}^{N}i(x_{j}.{\eta}_{j}+
v_{j}.{\xi}_{j}))dm_{t}^{(n)}({\xi}_{1},.. \
.,{\xi}_{n};{\eta}_{1},.. \ .,{\eta}_{n})\eqno(2.23)$$
{\it Moreover, $m_{t}^{(n)}$ is the restriction to
$(K{\times}{\bf
R}^{3})^{(n)}$ of a unique probability measure
$m_{t}$ on $(K{\times}{\bf R}^{3})^{\bf N}.$ }
\vskip 0.2cm\noindent
(B) {\it The initial form of $m$ is given by}
$$dm_{0}^{(n)}(x_{1},.. \ .,x_{n};v_{1},.. \ .,v_{n})=
{\Pi}_{j=1}^{n}dm_{0}^{(1)}(x_{j},v_{j})\eqno(2.24)$$
{\it where, as a consequence [3] of $(I.1)$ and $(I.3),$
$m_{0}^{(1)}$ is given formally by}
$$dm_{0}(x,v)={\sigma}_{0}(x){\delta}(v-u_{0}(x))$$
{\it i.e.}
$${\int}dm_{0}^{(1)}(x,v)f(x,v)={\int}dx{\sigma}_{0}(x)
f(x,u_{0}(x))\eqno(2.25)$$
{\it for test functions, $f,$ on $K{\times}{\bf R}^{3},$ that are
continuous and have compact support.}
\vskip 0.2cm\noindent
(C) {\it The probability measure $m_{t}$ evolves according to the
Vlasov hierarchy, in its weak form, i.e.}
$${d\over dt}{\int}f^{(n)}(x_{1},. \ .,x_{n};v_{1},. \ .,v_{n})
dm_{t}^{(n)}(x_{1},. \ .,x_{n};v_{1},. \ .,v_{n})=$$
$$ \ \ \ {\sum}_{j=1}^{n}{\int}v_{j}.{{\partial}f^{(n)}\over
{\partial}x_{j}}(x_{1},. \ .,x_{n};v_{1},. \ .,v_{n})
dm_{t}^{(n)}(x_{1},. \ .,x_{n};v_{1},. \ .,v_{n})$$
$$ \ \ \ -{\sum}_{j=1}^{n}{\int}{\nabla}U(x_{j}-x_{n+1}).
{{\partial}f^{(n)}\over {\partial}v_{j}}
(x_{1},. \ .,x_{n};v_{1},. \ .,v_{n})
dm_{t}^{(n+1)}(x_{1},. \ .,x_{n+1};v_{1},. \ .,v_{n+1})
\eqno(2.26)$$
{\it for test-functions $f^{(n)}$ that are continuously
differentiable and have compact support.} 
\vskip 0.2cm\noindent
(D) [3,5] {\it The decorrelation property (B) is
preserved
at all times, i.e.}
$$dm_{t}^{(n)}(x_{1},.. \ .,x_{n};v_{1},.. \ .,v_{n})=
{\Pi}_{j=1}^{n}dm_{t}^{(1)}(x_{j},v_{j})\eqno(2.27)$$
{\it and consequently, by (2.26), $m_{t}^{(1)},$ evolves
according to the weak form of the classical Vlasov equation,
i.e.}
$${d\over dt}{\int}f(x,v)dm_{t}^{(1)}(x,v)=
{\int}v.{{\partial}\over {\partial}x}f(x,v)dm_{t}^{(1)}(x,v)$$
$$ \ \ -{\int}{\nabla}U(x-y).{{\partial}\over {\partial}v}f(x,v)
dm_{t}^{(1)}(x,v)dm_{t}^{(1)}(y,w)\eqno(2.28)$$
\vskip 0.2cm\noindent
(E) [9] {\it This last equation has a unique solution, 
for the given initial conditions. Further, it is related to the 
(unique) solution of the Newtonian mean field theoretic problem}
$${d{\chi}_{t}(x,v)\over dt}={\cal V}_{t}(x,v); \ 
{d{\cal V}_{t}(x,v)\over dt}=
-{\int}dm_{0}(y,w)){\nabla}U({\chi}_{t}(x,v)-{\chi}_{t}(y,w));
\eqno(2.29)$$
{\it with}
$$ \ \ {\chi}_{0}(x,v)=x; \ {\cal V}_{0}(x,v)=v\eqno(2.30)$$
{\it by the formula}
$${\int}dm_{t}^{(1)}(x,v)f(x,v)={\int}dm_{0}^{(1)}(x,v)
f({\chi}_{t}(x,v),{\cal V}_{t}(x,v))\eqno(2.31)$$
{\it for test functions $f,$ that are continuous and have compact
support. Thus, defining}
$$X_{t}(x)={\chi}_{t}(x,u_{0}(x)); \ V_{t}(x)=
{\cal V}_{t}(x,u_{0}(x))\eqno(2.32)$$
{\it it follows from (2.25) that (2.29-31) may be re-expressed
as}
$${dX_{t}(x)\over dt}=V_{t}(x); \ {dV_{t}(x)\over dt}=
-{\int}dy{\sigma}_{0}(y){\nabla}U(X_{t}(x)-X_{t}(y)); \ 
X_{0}(x)=x; \ V_{0}(x)=u_{0}(x)\eqno(2.33)$$
{\it and}
$${\int}dm_{t}^{(1)}(x,v)f(x,v)={\int}dx{\sigma}_{0}(x)
f(X_{t}(x),V_{t}(x))\eqno(2.34)$$
\vskip 0.3cm\noindent
{\ssubt The Coulomb Model ${\Sigma}^{(N)}.$} The problems posed
by
this model stem from the singularity in the Coulomb potential,
$U_{c}.$ Our treatment of the model will be based on certain
regularity assumptions, designed to represent the idea that the
repulsive character of the inter-electronic forces tends to keep
the particles apart and thus tames the Coulomb singularity.
\vskip 0.2cm\noindent 
We express the first of these assumptions in terms of the one-
and two-particle spatial densities, ${\rho}^{(N,1)}, \
{\rho}^{(N,2)},$ specified by (2.16). We employ these densities
to define the conditional expectation, ${\cal
E}_{t}^{(N)}(f(x,y){\vert}x),$ of a two-point function $f(x,y),$
given $x,$ by the standard formula
$${\int}dx{\rho}_{t}^{(N,1)}(x){\cal E}_{t}^{(N)}(f(x,y){\vert}x)
g(x)={\int}dxdy{\rho}_{t}^{(N,2)}(x,y)f(x,y)g(x)\eqno(2.35)$$
for all continuous functions $g$ on $K.$ We then introduce the
following regularity assumption, to the effect that the Coulomb
repulsion keeps the electrons apart sufficiently to ensure that
the magnitude of the internal electric field remains bounded.
\vskip 0.2cm\noindent
{\it (R.1) For any finite ${\tau}(>0),$ there is a constant,
$B_{\tau}(<{\infty}),$ such that}
$${\int}dy{\rho}_{t}^{(N,1)}(x){\vert}{\nabla}U(x-y){\vert}
<B_{\tau}\eqno(2.36)$$
{\it and}
$${\cal E}_{t}^{(N)}({\vert}{\nabla}U(x-y){\vert} \ {\vert}x)
<B_{\tau}
 \ {\forall}t{\in}[0,{\tau}], \ x{\in}K\eqno(2.37)$$
\vskip 0.2cm\noindent
{\ssubt Comments.} (1) This is {\it implicitly} a condition on
the
initial state ${\psi}_{0}^{(N)}.$ It is indeed restrictive since
wave-functions can be constructed in such a way that their
evolution leads, in the limit $N{\rightarrow}{\infty},$ to a
catastrophic collapse, in which the microscopic dynamics breaks
down within a finite time [10].  
\vskip 0.2cm\noindent
(2) Assumption (R.1)) is weaker than the corresponding one of
Ref. [3], which required that ${\rho}_{t}^{(N,2)}$ itself was
uniformly bounded over finite time intervals and led to a smooth 
hydrodynamics from which dynamical phase transitions were
excluded. By contrast, (R.1) admits the possibility of
singularities in both ${\rho}_{t}^{(N,1)}$ and
${\rho}_{t}^{(N,2)},$ in the limit $N{\rightarrow}{\infty},$ and
thus, as we shall see, of hydrodynamical phase transitions.
\vskip 0.2cm\noindent
The following Proposition is an extension of results obtained in
Ref. [3] to the situation where the assumption of the uniform
boundedness of ${\rho}_{t}^{(N,2)}$ is replaced by (R.1). It is
a straightforward matter to check that the proofs given there of
these results prevail under the above weaker assumption. 
\vskip 0.3cm\noindent
{\ssubt Proposition 2.1.} {\it Assuming the conditions (I.1-4)
and (R.1), the above results (A)-(C) are valid for the Jellium
model, with the modification that here the convergence of
${\mu}_{t}^{(N,n)}$ to ${\mu}_{t}^{(n)}$ is subsequential. Thus,
the system evolves according to the classical Vlasov hierarchy
(2.26), subject to the initial conditions (2.24) and (2.25).
Furthermore, the magnitude of the electric field, in the limit
$N{\rightarrow}{\infty},$ satisfies the estimate}
$${\int}dm_{t}^{(1)}(y,w){\vert}U(x-y){\vert}<B_{\tau} \
{\forall}
x{\in}K, \ t{\in}[0,{\tau}]\eqno(2.38)$$
\vskip 0.3cm\noindent
We now assume that, as in ${\Sigma}_{g}^{(N)},$ the {\it
macroscopic} decorrelation property (I.4) persists in time, i.e.
that the (assumedly tamed) Coulomb singularity does not lead
to correlations of long range on the microscopic scale.
\vskip 0.2cm\noindent
{\it (R.2) The factorisation property (2.27) prevails at all
times.}
\vskip 0.2cm\noindent
As an immediate consequence of Prop. 2.1 and (R.2), we have
\vskip 0.3cm\noindent
{\ssubt Proposition 2.2.} {\it Under the further assumption
(R.2),
the single particle probability measure, $m_{t}^{(1)},$ evolves
according to the weak form (2.28) of the classical Vlasov
equation, subject to the initial condition (2.25).}
\vskip 0.3cm\noindent
Our next regularity assumption is that as the (assumedly tamed)
Coulomb singularity does not affect the uniqueness property (D),
that was operative for the regularised model. 
\vskip 0.2cm\noindent
{\it (R.3) The Vlasov equation (2.28), subject to the regularity
condition (2.38) and the given initial conditions, has a unique
solution.}
\vskip 0.2cm\noindent
Our last regularity condition is the counterpart to (R.3) for the
Newtonian mean field dynamics, as given by (2.33).
\vskip 0.2cm\noindent
{\it (R.4) The Newtonian mean field problem (2.33), subject to
the regularity condition}  
$${\int}dy{\sigma}_{0}(y){\vert}{\nabla}U(X_{t}(x)-
X_{t}(y)){\vert}<B_{\tau}, \ {\forall}x{\in}K, \ t{\in}[0,{\tau}]
\eqno(2.39)$$
{\it has a unique solution.}
\vskip 0.2cm\noindent 
The following Proposition is an immediate consequence of
Prop.2.2 and assumptions (R.3,4), since equations (2.33) and
(2.34) imply that $m_{t}^{(1)}$ satisfies the Vlasov equation
(2.28).
\vskip 0.3cm\noindent
{\ssubt Proposition 2.3.}{\it Under the further assumptions
(R.3,4),
the time-dependent macroscopic probability measure is given by
(2.34), with $(X_{t},V_{t})$ the unique solution of the Newtonian
problem (2.33) for the Jellium model.}
\medskip\noindent
{\subt 3. Eulerian Versus Stochastic Hydrodynamics.} 
\medskip\noindent
The Newtonian mean field theory, given by (2.33), corresponds to
a {\it Lagrangian} hydrodynamics, in which $X_{t}(x)$ and
$V_{t}(x)$ are the position and velocity, respectively, of a
'fluid particle'; and the Vlasov equation (2.28) is just the
Liouville equation representing its probabilistic description.
Our aim now is to investigate the conditions, both for the
Jellium model and its regularised version, under which the Vlasov
dynamics reduces to a deterministic Eulerian hydrodynamics. In
fact, we shall show that it does so, provided that $X_{t}$ is an
invertible function of position; and that otherwise it is
stochastic. Note here that the invertibility of the canonical
transformation
$(x,v){\rightarrow}({\chi}_{t}(x,v),{\cal V}_{t}(x,v))$ does {\it
not} imply that of the mapping
$x{\rightarrow}X_{t}(x){\equiv}{\chi}_{t}(x,u_{0}(x)).$
\vskip 0.2cm\noindent
{\bf Case (a):} $X_{t}$ {\bf Invertible.} In this case, the
Jacobean 
$$J_{t}(x)={{\partial}(X_{1,t},X_{2,t},X_{3,t})\over {\partial}
(x_{1},x_{2},x_{3})}\eqno(3.1)$$ 
with $x_{j}$ (resp. $X_{j,t}$) the j'th component $x$ (resp.
$X_{t}$), 
is strictly positive, and so we can define
$${\sigma}_{t}(x)={\sigma}_{0}(X_{t}^{-1}(x))/J_{t}(x)
\eqno(3.2)$$ 
and
$$u_{t}(x)=V_{t}(X_{t}^{-1}(x))\eqno(3.3)$$ 
Thus, since, by (2.34), (3.2) and (3.3),
$${\int}dm_{t}^{(1)}(x,v)f(x,v)={\int}dx{\sigma}_{t}(x)
f(x,u_{t}(x))\eqno(3.4)$$
for continuous functions $f$ on $K,$ i.e., formally,
$$dm_{t}(x,v)={\sigma}_{t}(x){\delta}(v-u_{t}(x))dxdv$$
it follows that $u_{t}(x)$ and ${\sigma}_{t}(x)$ are the drift
velocity and normalised particle density, respectively, at
position $x$ and time $t.$
\vskip 0.2cm\noindent
It follows now from (2.14), (2.22), (2.33), (3.2) and (3.3) 
that $u_{t},{\sigma}_{t}$ evolve according to the following
Euler-Maxwell hydrodynamical equations, previously obtained in
Ref. [3].
$${{\partial}{\sigma}_{t}\over {\partial}t}+{\nabla}.
({\sigma}_{t}u_{t})=0\eqno(3.5)$$
$${{\partial}u_{t}\over {\partial}t}+(u_{t}.{\nabla})u_{t}=E_{t}
\eqno(3.6)$$
where, for the Jellium model,
$${\nabla}.E_{t}=({\sigma}_{t}-1)\eqno(3.7)$$
and, for the regularised one,
$${\nabla}.E_{t}=({\sigma}_{t}^{(g)}-1)\eqno(3.7)^{\prime}$$
where
$${\sigma}_{t}^{(g)}(x)={\int}dyg(x-y){\sigma}_{t}(y)\eqno(3.8)$$
\vskip 0.3cm\noindent
{\bf Case (b): $X_{t}$ Non-Invertible.} In this case, we cannot
employ the formulae (3.2) and (3.3) to define the time-dependent
density and drift velocity. Instead, we have to consider the
situation where the equation
$$X_{t}(y)=x\eqno(3.9)$$
has several solutions, labelled by an index set $J,$ for $y$ as
a function of $x$ and $t,$ i.e.
$$y={\lbrace}Y_{t}^{(j)}(x){\vert}j{\in}J{\rbrace}\eqno(3.10)$$
In this case, equation (2.34) implies that
$${\int}dm_{t}^{(1)}(x,v)f(x,v)={\sum}_{j{\in}J}{\int}dx
{\sigma}_{t}^{(j)}(x)f(x,u_{t}^{(j)}(x))\eqno(3.11)$$
where
$${\sigma}_{t}^{(j)}(x)={\sigma}_{0}(Y_{t}^{(j)}(x)){\vert}
K_{t}^{(j)}(x){\vert}; \ \ u_{t}^{(j)}(x)=
V_{t}(Y_{t}^{(j)}(x))\eqno(3.12)$$
and
$$K_{t}^{(j)}(x)={{\partial}(Y_{1,t}^{(j)},Y_{2,t}^{(j)}),
Y_{3,t}^{(j)})\over {\partial}(x_{1},x_{2},x_{3})}\eqno(3.13)$$
Thus, (3.11) signifies that, formally,
$$dm_{t}(x,v)={\sum}_{j{\in}J}{\sigma}_{t}^{(j)}
{\delta}(v-u_{t}^{(j)}(x))dxdv$$
i.e. that the macroscopic state of the system at time t
corresponds to a statistical mixture of different streams, the
j'th of which has density ${\sigma}_{t}^{(j)}$ and drift velocity
$u_{t}^{(j)}.$ This implies that the local density and drift
velocity are now {\it stochastic} variables of a hydrodynamics
still governed by the Vlasov equation.
\vskip 0.2cm\noindent
We may summarise the above observations in the following form.
\vskip 0.3cm\noindent
{\ssubt Proposition 3.1.} {\it If $X_{t}$ is invertible, then the
macroscopic dynamics of the system corresponds to a hydrodynamics
given by the Euler-cum-Maxwell equations (3.5)-(3.7) (or
(3.7)$^{\prime}$). Otherwise, it corresponds to the flow of a
mixture of streams, and its evolution is of a stochastic type,
governed by the Vlasov equation (2.28).}
\vskip 0.3cm\noindent
{\ssubt Comment.} Unless the domain of non-invertibility of
$X_{t}$ is confined to a surface, the resultant mixture of
streams does not correspond to a shock wave. On this basis, it
will be seen that the example of Eulerian hydrodynamic breakdown
in ${\S}4$ is not that of a shock front.
\vfill\eject
\noindent 
{\subt 4. Example of Hydrodynamic Phase Transition.}  
\medskip\noindent
We shall now provide an example of initial conditions, which lead
to a transition from a deterministic to a stochastic flow, in
both the regularised and the Coulomb models. These are conditions
where both ${\sigma}_{0}$ and $u_{0}$ are functions of just a
single coordinate, say $x_{1},$ and $u_{0}$ is directed along
$Ox_{1}.$ In this case, the flow becomes effectively one-
dimensional, for the following reasons. If 
$$X_{t}^{(b)}(x):=X_{t}(x+b)-b$$
for arbitrary vectors $b$ in the plane $Ox_{2}x_{3},$ then, in
view of the periodicity of $K$, if $X_{t}$ is a solution of the
Newtonian problem (2.33), so too is $X_{t}^{(b)}.$ Hence, by the
uniqueness\footnote *{In the case of the
Coulomb model, this is a consequence of assumption (R.4).} of the
solution of (2.33), $X_{t}{\equiv}X_{t}^{(b)},$ which implies
that the component $X_{1,t}$ of $X_{t}$ depends on the coordinate
$x_{1}$ only, and that $X_{2,t}(x),X_{3,t}(x)$ reduce to
$x_{2}+{\xi}_{2}(t), \ x_{3}+{\xi}_{3}(t),$ where the ${\xi}'$s 
are functions of $t$ only. Furthermore, it follows from the
$x_{2}-$ and $x_{3}-$components of (2.33) that these functions
are both zero. In other words, the component $X_{1,t}$ of $X_{t}$
depends only on $x_{1},$ while $X_{2,t},X_{3,t}$ remain fixed at
$x_{2},x_{3},$ respectively. Hence, the macroscopic dynamics
reduces to a one-dimensional flow. For notational convenience,
we shall henceforth drop the suffix $1$ from $X_{1,t}$ and
$x_{1}.$  
\vskip 0.2cm\noindent
Thus, by (2.33),
$$X_{t}(x)=x+u_{0}(x)t+{\int}_{0}^{t}ds(t-s){\int}_{0}^{1}dy
{\sigma}_{0}(y)F(X_{s}(x)-X_{s}(y))\eqno(4.1)$$
where
$$F(x)=-{\int}_{0}^{1}dx_{2}{\int}_{0}^{1}dx_{3}
{{\partial}U\over {\partial}x}(x,x_{2},x_{3})\eqno(4.2)$$
Let
$$J_{t}(x)={{\partial}X_{t}(x)\over {\partial}x}\eqno(4.3)$$
Then, by Prop. 3.1 and the implicit function theorem, the
neccessary and sufficient condition for deterministic
hydrodynamics is that $J_{t}$ has no zeroes. We note also that
the definition (4.3) permits us to re-express (4.1) in the form
$$X_{t}(x)=x+u_{0}(x)t+{\int}_{0}^{t}ds(t-s){\int}_{0}^{1}dy
{\sigma}_{0}(y)F({\int}_{y}^{x}dzJ_{s}(z)))
\eqno(4.1)^{\prime}$$
\vskip 0.3cm\noindent
{\ssubt The Regularised Model.} Here, $U=U_{g}$ and thus, by
(4.2), $F$ is a continuously differentiable function. Hence, by
(4.1) and (4.3),
$$J_{t}(x)(1-{\int}_{0}^{t}ds(t-s){\int}_{0}^{1}dy{\sigma}_{0}(y)
F^{\prime}(x-y))=1+u_{0}^{\prime}(x)t\eqno(4.4)$$
where the primes denote differentiation w.r.t. $x.$ Thus,
defining 
$${\Vert}F^{\prime}{\Vert}=sup{\lbrace}{\vert}F^{\prime}
(x){\vert} \ {\vert}x{\in}[0,1]{\rbrace},$$ 
and $t_{0},t_{1}$ to be the times
given by
$$t_{0}={2/{\Vert}F^{\prime}{\Vert}}^{1/2}\eqno(4.5)$$ 
and
$$t_{1}=min{\lbrace}t(>0){\vert} \ (1+u_{0}^{\prime}(x)t)=0 \ for
\ some \ x{\in}[0,1]{\rbrace}\eqno(4.6)$$
it follows immediately from (4.4)-(4.6) that $J_{t}$ is
strictly positive, and hence that $X_{t}$ is invertible, if
$0{\le}t<min(t_{0},t_{1}).$ Therefore, by Prop. 3.1, the system
evolves, in this regime, according to a deterministic
hydrodynamics, given by equations (3.5), (3.6), (3.7)$^{\prime}$
and (3.8).
\vskip 0.2cm\noindent
On the other hand, if the initial conditions are such that
$t_{1}<t_{0},$ then it follows from (4.4)-(4.6) that $J_{t}$
changes sign during the interval $t{\in}(t_{1},t_{0})$ over some
spatial domain $D({\subset}[0,1]).$ Hence, by Prop. 3.1, the
hydrodynamics of the model becomes stochastic, and is (still)
governed by the Vlasov equation (2.28).
\vskip 0.2cm\noindent
We may summarise these results as follows.
\vskip 0.3cm\noindent
{\ssubt Proposition 4.1.} {\it The regularised model exhibits a
deterministic hydrodynamics, given by equations (3.5), (3.6) and
(3.7)$^{\prime}$, over the time interval
$0{\le}t<min(t_{0},t_{1}).$
However, if the initial velocity profile is such that
$t_{1}<t_{0},$ then the flow undergoes a transition to
stochasticity at time $t_{1}.$} 
\vskip 0.3cm\noindent
{\ssubt Comment.} It will be seen from the derivations of this
result that, in the stochastic phase, the domain of
non-invertibility of $X_{t}$ is, in general, not confined to a
single value of $x,$ i.e. to a surface in $K.$ Thus, in view of
the comment following Prop. 3.1, the hydrodynamic phase
transition described here does not correspond to the formation
of a shock wave.
\vskip 0.3cm\noindent
{\ssubt The Coulomb Model.} We shall prove the following
Proposition for this model.
\vskip 0.3cm\noindent
{\ssubt Proposition 4.2.} {\it Under the specified assumptions,
the hydrodynamics of the Coulomb model takes the deterministic
form (3.5)-(3.7) at all times, provided that the initial density
and velocity profiles satisfy the condition}
$$({\sigma}_{0}(x)-1)^{2}+(u_{0}^{\prime}(x))^{2}<({\sigma}_{0}
(x))^{2} \ {\forall}x{\in}[0,1]\eqno(4.7)$$
{\it Otherwise there is a transition to a stochastic flow at a
certain time ${\tau},$ given by the least positive value of $t$
for which} 
$${\sigma}_{0}(x)+(1-{\sigma}_{0}(x))
{\cos}(t)+u_{0}^{\prime}(x){\sin}(t)=0\eqno(4.8)$$
{\it for some $x{\in}[0,1].$}
\vskip 0.3cm\noindent
{\ssubt Comment.} Again, the domain of non-invertibility of
$X_{t}$ in the stochastic phase is not confined to a single value
of $x,$ i.e. to a surface in $K;$ and thus the hydrodynamical
phase transition does not correspond to the formation of a shock
wave.
\vskip 0.3cm\noindent
To prove this Proposition, we shall first establish the following
lemma.
\vskip 0.3cm\noindent
{\ssubt Lemma 4.3.} {\it If $X_{t}$ is invertible for
$t{\in}[0,{\tau}],$ where ${\tau}>0,$ then the regularity
condition (2.39) is satisfied.}
\vskip 0.3cm\noindent
{\ssubt Proof.} We note first that, by (2.13) and (4.2),
$$F(x)={\sum}_{n=1}^{\infty}i{\exp}
(2{\pi}inx)/(2{\pi}n)$$
which implies that $F$ is square integrable, hence absolutely
integrable, over $[0,1].$ Thus, since, by
(3.2), (4.2) and (4.3), the l.h.s. of (2.39) is equal to
$${\int}_{0}^{1}dy{\sigma}_{0}(y)J_{t}(y){\vert}F
(X_{t}(x)-X_{t}(y)){\vert}$$
$${\equiv}{\int}_{0}^{1}dy{\sigma}_{0}(X_{t}^{-1}(y))
{\vert}F(X_{t}(x)-y){\vert}$$
by the invertibility of $X_{t},$ it follows that the condition
(2.39) is satisfied.
\vskip 0.3cm\noindent
{\ssubt Proof of Prop. 4.2.} Since $U=U_{c}$ here, it follows
from (2.13), (2.14), (4.1) and (4.3) that
$$J_{t}(x)=1+u_{0}^{\prime}(x)t+{\int}_{0}^{t}ds(t-s)
J_{s}(x)(-1+{\int}_{0}^{1}dy{\sigma}_{0}(y)
{\delta}(X_{s}(x)-X_{s}(y)))\eqno(4.9)$$
where now ${\delta}$ is the Dirac distribution on $[0,1],$
subject to periodic boundary conditions. 
\vskip 0.2cm\noindent
Let us first suppose that $X_{t}$ is invertible, i.e. that
$J_{t}$ is strictly positive, over a time interval
$0{\le}t<{\tau}_{0},$ for some positive ${\tau}_{0}.$ In this
case, 
$$J_{s}(x){\delta}(X_{s}(x)-X_{s}(y)){\equiv}{\delta}(x-y) \
{\forall}s{\in}[0,{\tau}_{0})$$
and therefore (4.9) reduces to
$$J_{t}(x)=1+u_{0}^{\prime}(x)t+{\int}_{0}^{t}ds(t-s)
({\sigma}_{0}(x)-J_{s}(x))\eqno(4.10)$$
i.e.
$$({d^{2}\over dt^{2}}+1)J_{t}(x)={\sigma}_{0}(x);
 \ with \ J_{0}(x)=1; \ {\dot J}_{0}(x)
=u_{0}^{\prime}(x)\eqno(4.11)$$
where ${\dot J}_{t}=dJ_{t}/dt.$ Hence,
$$J_{t}(x)={\sigma}_{0}(x)+(1-{\sigma}_{0}(x))
{\cos}(t)+u_{0}^{\prime}(x){\sin}(t)\eqno(4.12)$$
In view of the non-negativity of ${\sigma}_{0},$ this equation
implies that $J_{t}$ is strictly positive for all $t{\ge}0$
if and only if the condition (4.7) is fulfilled. Otherwise,
$J_{t}$ changes sign at some point $x{\in}[0,1]$ when $t$ reaches
the value ${\tau}$ specified in the statement of the Proposition.
\vskip 0.2cm\noindent
We may thus summarise these results as follows.
\vskip 0.2cm\noindent
(a) If (4.7) is satisfied, then the function $X_{t}$ given by
substituting the formula (4.12) for $J_{t}$ into the r.h.s. of
(4.1)$^{\prime}$ is invertible and satisfies both the Newtonian
mean field equation (4.10) and, by Lemma (4.3), the regularity
condition (2.39), for all $t{\ge}0.$ Hence, by (R.4), it is the
unique solution of the Newtonian mean field equation, and
persists for all positive $t.$ Hence, by Prop. 3.1, the model
exhibits the Eulerian hydrodynamics given by (3.5)-(3.7) at all
times.
\vskip 0.2cm\noindent
(b) If (4.7) is violated, then, by the same argument, the
system exhibits this deterministic hydrodynamics for times
$t{\in}[0,{\tau}),$ with ${\tau}$ as specified in Prop. 4.2. 
\vskip 0.2cm\noindent
(c) If (4.7) is violated, then there must be a transition to
stochastic flow at $t={\tau},$ since an assumption to the
contrary becomes invalid when $t$ passes through that
value.
\vskip 0.2cm\noindent
The results (a)-(c) establish the Proposition.
\medskip\noindent
{\subt 5. Concluding Remarks.} 
\medskip\noindent
We have shown here that the quantum dynamics of the Jellium model
leads to a hydrodynamics, which supports both deterministic and
stochastic flows, and exhibits phase transitions between them. This 
hydrodynamics is therefore richer than that of the deterministic flow 
given by the Euler-cum-Maxwell equations. Furthermore, since the flow 
in the stochastic phase corresponds to a statistical mixture
of different streams, one might envisage that this carries
a germ of turbulence.
\vskip 0.2cm\noindent
As regards possible ramifications of the present work, we note 
that the above hydrodynamical properties of the model stemmed from 
its Vlasov dynamics, which is simply the Liouville probabilistic
version of its Lagrangian hydrodynamics (cf. ${\S}3$). This suggests 
that, more generally, a natural way of formulating the theory of 
stochastic flows, even of turbulence, might be via a probabilistic 
treatment of Lagrangian hydrodynamics. That should presumably have 
some connection with Foias's [11] formulation of stochastic 
hydrodynamics on the basis of a Liouville equation governing 
Navier-Stokes flows. It need not, however, be equivalent to it, 
since, as we have seen in ${\S}'s$ 3 and 4, the Eulerian and 
Lagrangian pictures of the present plasma model are not equivalent.
\vskip 0.2cm\noindent
Finally, we remark here that the hydrodynamics obtained here is
completely inviscid. The reason for this, as in Ref. [3] (cf.
discussion there at the end of ${\S}$1), is that our macroscopic
description is effected on the largest available length scale,
$L,$ and that, consequently, the viscous forces are 'scaled
away'. Thus, the hydrodynamic picture we have obtained should be
regarded as no more than a skeletal version of that of a real
plasma.
\medskip\noindent
{\subt References.}
\medskip\noindent
1. E. H. Lieb and H. Narnhofer: J. Stat. Phys. {\bf 12}, 291
(1975)
\vskip 0.2cm\noindent
2. Ph. Martin and Ch. Oguey: I. Phys. A {\bf 18}, 1995 (1985)
\vskip 0.2cm\noindent
3. G. L. Sewell: J. Math. Phys. {\bf 26}, 2324 (1985)
\vskip 0.2cm\noindent
4. E. B. Davies: J. Stat. Phys. {\bf 18}, 161 (1978)
\vskip 0.2cm\noindent
5. H. Narnhofer and G. L. Sewell: Commun. Math. Phys. {\bf 79},
9 (1981)
\vskip 0.2cm\noindent
6. D. Ruelle: "Statistical Mechanics", Benjamin, New York (1969)
\vskip 0.2cm\noindent
7. G. L. Sewell: "Quantum Theory of Collective Phenomena", 
Clarendon Press, Oxford (1991)
\vskip 0.2cm\noindent
8. H. Spohn: Math. Meth. Appl. Sci. {\bf 3}, 445 (1981)  
\vskip 0.2cm\noindent
9. H. Neunzert: Fluid Dyn. Trans. {\bf 9}, 229 (1978)
\vskip 0.2cm\noindent
10. C. Radin: Commun. Math. Phys. {\bf 54}, 69 (1974)
\vskip 0.2cm\noindent
11. C. Foias: Russian Math. Surveys {\bf 29}, 293 (1974)